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**Published in:** LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

We study the kernelization of exploration problems on temporal graphs. A temporal graph consists of a finite sequence of snapshot graphs 𝒢 = (G₁, G₂, … , G_L) that share a common vertex set but might have different edge sets. The non-strict temporal exploration problem (NS-TEXP for short) introduced by Erlebach and Spooner, asks if a single agent can visit all vertices of a given temporal graph where the edges traversed by the agent are present in non-strict monotonous time steps, i.e., the agent can move along the edges of a snapshot graph with infinite speed. The exploration must at the latest be completed in the last snapshot graph. The optimization variant of this problem is the k-arb NS-TEXP problem, where the agent’s task is to visit at least k vertices of the temporal graph. We show that under standard computational complexity assumptions, neither of the problems NS-TEXP nor k-arb NS-TEXP allow for polynomial kernels in the standard parameters: number of vertices n, lifetime L, number of vertices to visit k, and maximal number of connected components per time step γ; as well as in the combined parameters L+k, L + γ, and k+γ. On the way to establishing these lower bounds, we answer a couple of questions left open by Erlebach and Spooner.
We also initiate the study of structural kernelization by identifying a new parameter of a temporal graph p(𝒢) = ∑_{i=1}^L (|E(G_i)|) - |V(G)| + 1. Informally, this parameter measures how dynamic the temporal graph is. Our main algorithmic result is the construction of a polynomial (in p(𝒢)) kernel for the more general Weighted k-arb NS-TEXP problem, where weights are assigned to the vertices and the task is to find a temporal walk of weight at least k.

Emmanuel Arrighi, Fedor V. Fomin, Petr A. Golovach, and Petra Wolf. Kernelizing Temporal Exploration Problems. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 1:1-1:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{arrighi_et_al:LIPIcs.IPEC.2023.1, author = {Arrighi, Emmanuel and Fomin, Fedor V. and Golovach, Petr A. and Wolf, Petra}, title = {{Kernelizing Temporal Exploration Problems}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {1:1--1:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara and Wahlstr\"{o}m, Magnus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.1}, URN = {urn:nbn:de:0030-drops-194201}, doi = {10.4230/LIPIcs.IPEC.2023.1}, annote = {Keywords: Temporal graph, temporal exploration, computational complexity, kernel} }

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**Published in:** LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

Cluster Editing, also known as correlation clustering, is a well-studied graph modification problem. In this problem, one is given a graph and allowed to perform up to k edge additions and deletions to transform it into a cluster graph, i.e., a graph consisting of a disjoint union of cliques. However, in real-world networks, clusters are often overlapping. For example, in social networks, a person might belong to several communities - e.g. those corresponding to work, school, or neighborhood. Another strong motivation comes from language networks where trying to cluster words with similar usage can be confounded by homonyms, that is, words with multiple meanings like "bat". The recently introduced operation of vertex splitting is one natural approach to incorporating such overlap into Cluster Editing. First used in the context of graph drawing, this operation allows a vertex v to be replaced by two vertices whose combined neighborhood is the neighborhood of v (and thus v can belong to more than one cluster). The problem of transforming a graph into a cluster graph using at most k edge additions, edge deletions, or vertex splits is called Cluster Editing with Vertex Splitting and is known to admit a polynomial kernel with respect to k and an O(9^{k²} + n + m)-time (parameterized) algorithm. However, it was not known whether the problem is NP-hard, a question which was originally asked by Abu-Khzam et al. [Combinatorial Optimization, 2018]. We answer this in the affirmative. We further give an improved algorithm running in O(2^{7klog k} + n + m) time.

Emmanuel Arrighi, Matthias Bentert, Pål Grønås Drange, Blair D. Sullivan, and Petra Wolf. Cluster Editing with Overlapping Communities. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 2:1-2:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{arrighi_et_al:LIPIcs.IPEC.2023.2, author = {Arrighi, Emmanuel and Bentert, Matthias and Drange, P\r{a}l Gr{\o}n\r{a}s and Sullivan, Blair D. and Wolf, Petra}, title = {{Cluster Editing with Overlapping Communities}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {2:1--2:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara and Wahlstr\"{o}m, Magnus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.2}, URN = {urn:nbn:de:0030-drops-194218}, doi = {10.4230/LIPIcs.IPEC.2023.2}, annote = {Keywords: graph modification, correlation clustering, vertex splitting, NP-hardness, parameterized algorithm} }

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PACE Solver Description

**Published in:** LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

The graph parameter twin-width was recently introduced by Bonnet et al. Twin-width is a parameter that measures a graph’s similarity to a cograph, which is a graph that can be reduced to a single vertex by repeatedly contracting twins. This brief description introduces Zygosity, a heuristic for computing a low-width contraction sequence that achieved second place in the 2023 edition of Parameterized Algorithms and Computational Experiments Challenge (PACE). Zygosity starts by repeatedly contracting twins. Then, any attached trees are contracted down to a single pendant vertex. The remaining graph is then contracted using a randomized greedy algorithm.

Emmanuel Arrighi, Pål Grønås Drange, Kenneth Langedal, Farhad Vadiee, Martin Vatshelle, and Petra Wolf. PACE Solver Description: Zygosity. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 39:1-39:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{arrighi_et_al:LIPIcs.IPEC.2023.39, author = {Arrighi, Emmanuel and Drange, P\r{a}l Gr{\o}n\r{a}s and Langedal, Kenneth and Vadiee, Farhad and Vatshelle, Martin and Wolf, Petra}, title = {{PACE Solver Description: Zygosity}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {39:1--39:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara and Wahlstr\"{o}m, Magnus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.39}, URN = {urn:nbn:de:0030-drops-194583}, doi = {10.4230/LIPIcs.IPEC.2023.39}, annote = {Keywords: Twin-width, randomized greedy algorithm} }

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**Published in:** LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

In the Intersection Non-emptiness problem, we are given a list of finite automata A_1, A_2,… , A_m over a common alphabet Σ as input, and the goal is to determine whether some string w ∈ Σ^* lies in the intersection of the languages accepted by the automata in the list. We analyze the complexity of the Intersection Non-emptiness problem under the promise that all input automata accept a language in some level of the dot-depth hierarchy, or some level of the Straubing-Thérien hierarchy. Automata accepting languages from the lowest levels of these hierarchies arise naturally in the context of model checking. We identify a dichotomy in the dot-depth hierarchy by showing that the problem is already NP-complete when all input automata accept languages of the levels B_0 or B_{1/2} and already PSPACE-hard when all automata accept a language from the level B_1. Conversely, we identify a tetrachotomy in the Straubing-Thérien hierarchy. More precisely, we show that the problem is in AC^0 when restricted to level L_0; complete for L or NL, depending on the input representation, when restricted to languages in the level L_{1/2}; NP-complete when the input is given as DFAs accepting a language in L_1 or L_{3/2}; and finally, PSPACE-complete when the input automata accept languages in level L_2 or higher. Moreover, we show that the proof technique used to show containment in NP for DFAs accepting languages in L_1 or L_{3/2} does not generalize to the context of NFAs. To prove this, we identify a family of languages that provide an exponential separation between the state complexity of general NFAs and that of partially ordered NFAs. To the best of our knowledge, this is the first superpolynomial separation between these two models of computation.

Emmanuel Arrighi, Henning Fernau, Stefan Hoffmann, Markus Holzer, Ismaël Jecker, Mateus de Oliveira Oliveira, and Petra Wolf. On the Complexity of Intersection Non-emptiness for Star-Free Language Classes. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{arrighi_et_al:LIPIcs.FSTTCS.2021.34, author = {Arrighi, Emmanuel and Fernau, Henning and Hoffmann, Stefan and Holzer, Markus and Jecker, Isma\"{e}l and de Oliveira Oliveira, Mateus and Wolf, Petra}, title = {{On the Complexity of Intersection Non-emptiness for Star-Free Language Classes}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {34:1--34:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-215-0}, ISSN = {1868-8969}, year = {2021}, volume = {213}, editor = {Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.34}, URN = {urn:nbn:de:0030-drops-155456}, doi = {10.4230/LIPIcs.FSTTCS.2021.34}, annote = {Keywords: Intersection Non-emptiness Problem, Star-Free Languages, Straubing-Th\'{e}rien Hierarchy, dot-depth Hierarchy, Commutative Languages, Complexity} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

In this work, we consider the following order reconfiguration problem: Given a graph G together with linear orders ω and ω' of the vertices of G, can one transform ω into ω' by a sequence of swaps of adjacent elements in such a way that at each time step the resulting linear order has cutwidth (pathwidth) at most k? We show that this problem always has an affirmative answer when the input linear orders ω and ω' have cutwidth (pathwidth) at most k/2. Using this result, we establish a connection between two apparently unrelated problems: the reachability problem for two-letter string rewriting systems and the graph isomorphism problem for graphs of bounded cutwidth. This opens an avenue for the study of the famous graph isomorphism problem using techniques from term rewriting theory.

Emmanuel Arrighi, Henning Fernau, Mateus de Oliveira Oliveira, and Petra Wolf. Order Reconfiguration Under Width Constraints. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{arrighi_et_al:LIPIcs.MFCS.2021.8, author = {Arrighi, Emmanuel and Fernau, Henning and de Oliveira Oliveira, Mateus and Wolf, Petra}, title = {{Order Reconfiguration Under Width Constraints}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {8:1--8:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.8}, URN = {urn:nbn:de:0030-drops-144486}, doi = {10.4230/LIPIcs.MFCS.2021.8}, annote = {Keywords: Parameterized Complexity, Order Reconfiguration, String Rewriting Systems} }

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**Published in:** LIPIcs, Volume 190, 19th International Symposium on Experimental Algorithms (SEA 2021)

In the Steiner tree problem, the input consists of an edge-weighted graph G together with a set S of terminal vertices. The goal is to find a minimum weight tree in G that spans all terminals. This fundamental NP-hard problem has direct applications in many subfields of combinatorial optimization, such as planning, scheduling, etc. In this work we introduce a new heuristic for the Steiner tree problem, based on a simple routine for improving the cost of sub-optimal Steiner trees: first, the sub-optimal tree is split into three connected components, and then these components are reconnected by using an algorithm that computes an optimal Steiner tree with 3-terminals (the roots of the three components). We have implemented our heuristic into a solver and compared it with several state-of-the-art solvers on well-known data sets. Our solver performs very well across all the data sets, and outperforms most of the other benchmarked solvers on very large graphs, which have been either obtained from real-world applications or from randomly generated data sets.

Emmanuel Arrighi and Mateus de Oliveira Oliveira. Three Is Enough for Steiner Trees. In 19th International Symposium on Experimental Algorithms (SEA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 190, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{arrighi_et_al:LIPIcs.SEA.2021.5, author = {Arrighi, Emmanuel and de Oliveira Oliveira, Mateus}, title = {{Three Is Enough for Steiner Trees}}, booktitle = {19th International Symposium on Experimental Algorithms (SEA 2021)}, pages = {5:1--5:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-185-6}, ISSN = {1868-8969}, year = {2021}, volume = {190}, editor = {Coudert, David and Natale, Emanuele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2021.5}, URN = {urn:nbn:de:0030-drops-137775}, doi = {10.4230/LIPIcs.SEA.2021.5}, annote = {Keywords: Steiner Tree, Heuristics, 3TST} }

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**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

We are studying a weighted version of a linear extension problem, given some finite partial order ρ, called Completion of an Ordering. While this problem is NP-complete, we show that it lies in FPT when parameterized by the interval width of ρ. This ordering problem can be used to model several ordering problems stemming from diverse application areas, such as graph drawing, computational social choice, or computer memory management. Each application yields a special ρ. We also relate the interval width of ρ to parameterizations such as maximum range that have been introduced earlier in these applications, sometimes improving on parameterized algorithms that have been developed for these parameterizations before. This approach also gives some practical sub-exponential time algorithms for ordering problems.

Emmanuel Arrighi, Henning Fernau, Mateus de Oliveira Oliveira, and Petra Wolf. Width Notions for Ordering-Related Problems. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{arrighi_et_al:LIPIcs.FSTTCS.2020.9, author = {Arrighi, Emmanuel and Fernau, Henning and de Oliveira Oliveira, Mateus and Wolf, Petra}, title = {{Width Notions for Ordering-Related Problems}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {9:1--9:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.9}, URN = {urn:nbn:de:0030-drops-132505}, doi = {10.4230/LIPIcs.FSTTCS.2020.9}, annote = {Keywords: Parameterized algorithms, interval width, linear extension, one-sided crossing minimization, Kemeny rank aggregation, grouping by swapping} }

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